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Econometrics I
Professor William Greene
Stern School of Business
Department
of Economics
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Econometrics I
Part
15
–
Generalized
Regression
Applications
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Leading Applications of the GR Model
Heteroscedasticity and Weighted
Least Squares
Autocorrelation in Time Series Models
SUR Models for Production and Cost
VAR models in Macroeconomics and
Finance
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Two Step Estimation of the Generalized
Regression Model
Use the Aitken (Generalized Least Squares

GLS)
estimator with an estimate of
1.
i猠sarame瑥riedbyafee獴業ableparame瑥r献s
Examples, the heteroscedastic model
2. Use least squares residuals to estimate the variance
functions
3. Use the estimated
in䝌S

Fea獩ble䝌SⰠorF䝌S
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General Result for Estimation
When
䥳s瑩m慴ad
True GLS uses [
X

1
X
]
X

1
y
which
converges in probability to
.
We seek a vector which converges to the same
thing that this does. Call it “feasible” GLS,
FGLS, based on
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FGLS
Feasible GLS is based on finding an estimator
which has the same properties as the true GLS.
Example Var[
i
] =
2
[Exp(
z
i
)]2.
True GLS would regress y/[
Exp(
z
i
)]
on the same
transformation of
x
i
.
With a consistent estimator of [
,
崬獡y孳,
c
], we do
the same computation with our estimates.
So long as plim [s,
c
] = [
,
崬F䝌Si猠s猠
“good”
as
true GLS
.
Consistent
Same Asymptotic Variance
Same Asymptotic Normal Distribution
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FGLS vs. Full GLS
VVIR
(Theorem 9.6)
To achieve full efficiency, we do not
need an efficient estimate of the
parameters in
,onlyaonsistent
one.
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Heteroscedasticity
Setting:
The regression disturbances have unequal variances, but are
still not correlated with each other:
Classical regression with hetero

(different) scedastic (variance)
disturbances.
y
i
=
x
i
+
i
, E[
i
] = 0, Var[
i
] =
2
i
,
i
> 0.
The classical model arises if
i
= 1.
A normalization:
i
i
= n. Not a restriction, just a scaling that is
absorbed into
2
.
A characterization of the heteroscedasticity: Well defined estimators
and methods for testing hypotheses will be obtainable if the
heteroscedasticity is “well behaved” in the sense that no single
observation becomes dominant.
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Behavior of OLS
Implications for conventional estimation technique
and hypothesis testing:
1
. b
is still unbiased. Proof of unbiasedness did
not rely on homoscedasticity
2. Consistent? We need the more general proof.
Not difficult.
3. If plim
b
=
Ⱐ瑨enplim
2
=
2
(with the
normalization).
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Inference Based on OLS
What of s
2
(
X
X
)

1
? Depends on
X
堠

X
X
. If they are
nearly the same, the OLS covariance matrix is OK.
When will they be nearly the same? Relates to an
interesting property of weighted averages. Suppose
i
is randomly drawn from a distribution with E[
i
] = 1.
Then, (1/n)
i
x
i
2
E[x
2
] and (1/n)
i
i
x
i
2
E[x
2
].
This is the crux of the discussion in your text.
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Inference Based on OLS
VIR
: For the heteroscedasticity to be substantive wrt estimation and
inference by LS, the weights must be correlated with x and/or x
2
.
(Text, page 272.)
If the heteroscedasticity is important. Then,
b
is inefficient.
The White estimator.
ROBUST
estimation of the variance of
b
.
Implication for testing hypotheses. We will use Wald tests. Why?
(
ROBUST TEST STATISTICS
)
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Finding Heteroscedasticity
The central issue is whether E[
2
] =
2
i
is related
to the xs or their squares in the model.
Suggests an obvious strategy. Use residuals to
estimate disturbances and look for relationships
between e
i
2
and x
i
and/or x
i
2
. For example,
regressions of squared residuals on xs and their
squares.
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Procedures
White’s general test
: nR
2
in the regression of e
i
2
on all
unique xs, squares, and cross products. Chi

squared[P]
Breusch and Pagan’s Lagrange multiplier test
. Regress
{[e
i
2
/(
e
e
/n)]
–
1} on
Z
(may be
X
). Chi

squared. Is nR
2
with degrees of freedom rank of
Z
. (Very elegant.)
Others described in text for other purposes. E.g.,
groupwise heteroscedasticity. Wald, LM, and LR tests
all examine the dispersion of group specific least
squares residual variances.
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Estimation: WLS form of GLS
General result

mechanics of weighted least squares.
Generalized least squares

efficient estimation.
Assuming
weights are known.
Two step generalized least squares:
Step 1: Use least squares, then the residuals to
estimate the weights.
Step 2: Weighted least squares using the estimated
weights.
(Iteration: After step 2, recompute residuals and return to
step 1. Exit when coefficient vector stops changing.)
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Autocorrelation
The analysis of “autocorrelation” in the narrow sense of correlation
of the disturbances across time largely parallels the discussions
we’ve already done for the GR model in general and for
heteroscedasticity in particular. One difference is that the relatively
crisp results for the model of heteroscedasticity are replaced with
relatively fuzzy, somewhat imprecise results here. The reason is
that it is much more difficult to characterize meaningfully “well
behaved” data in a time series context. Thus, for example, in
contrast to the sharp result that produces the White robust
estimator, the theory underlying the Newey

West robust estimator is
somewhat ambiguous in its requirement of a bland statement about
“how far one must go back in time until correlation becomes
unimportant.”
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The Familiar AR(1) Model
t
=
t

1
+
u
t
, 
 <
1
.
This characterizes the disturbances, not the regressors.
A general characterization of the mechanism producing
history + current innovations
Analysis of this model in particular. The mean and variance
and autocovariance
Stationarity. Time series analysis.
Implication: The form of
2
;
Var[
] vs. Var[u].
Other models for autocorrelation

less frequently used
–
AR(1) is the workhorse.
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Building the Model
Prior view: A feature of the data
“Account for autocorrelation in the data.”
Different models, different estimators
Contemporary view: Why is there autocorrelation?
What is missing from the model?
Build in appropriate dynamic structures
Autocorrelation should be “built out” of the model
Use robust procedures (Newey

West) instead of elaborate
models specifically for the autocorrelation.
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Model Misspecification
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Implications for Least Squares
Familiar results: Consistent, unbiased, inefficient, asymptotic normality
The inefficiency of least squares:
Difficult to characterize generally. It is worst in “low frequency”
i.e., long period (year) slowly evolving data.
Can be extremely bad. GLS vs. OLS, the efficiency ratios can
be 3 or more.
A very important exception

the lagged dependent variable
y
t
=
x
t
+
y
t

1
+
t
.
t
=
t

1
+ u
t
,.
Obviously, Cov[y
t

1
,
t
]
0, because of the form of
t
.
How to estimate? IV
Should the model be fit in this form? Is something missing?
Robust estimation of the covariance matrix

the Newey

West
estimator.
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GLS and FGLS
Theoretical result for known

i⸬no睮
.
Prais

Winsten vs. Cochrane

Orcutt.
FGLS estimation: How to estimate
? OLS
residuals as usual

first autocorrelation.
Many variations, all based on correlation of e
t
and
e
t

1
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Testing for Autocorrelation
A general proposition: There are several tests. All are functions of the
simple autocorrelation of the least squares residuals. Two used
generally, Durbin

Watson and Lagrange Multiplier
The Durbin

Watson test. d
2(1

r). Small values of d lead to
rejection of
NO AUTOCORRELATION: Why are the bounds necessary?
Godfrey’s LM test. Regression of e
t
on e
t

1
and
x
t
. Uses a “partial
correlation.”
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Consumption “Function”
Log real consumption vs. Log real disposable income
(
Aggregate U.S. Data, 1950I
–
2000IV. Table F5.2 from text)

Ordinary least squares regression ............
LHS=LOGC Mean = 7.88005
Standard deviation = .51572
Number of observs. = 204
Model size Parameters = 2
Degrees of freedom = 202
Residuals Sum of squares = .09521
Standard error of e = .02171
Fit R

squared = .
99824 <<<***
Adjusted R

squared = .99823
Model test F[ 1, 202] (prob) =114351.2(.0000)

+

Variable Coefficient Standard Error t

ratio P[T>t] Mean of X

+

Constant

.13526*** .02375

5.695 .0000
LOGY 1.00306*** .00297 338.159 .0000 7.99083

+

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Least Squares Residuals: r = .91
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Conventional vs. Newey

West
+

+

+

+

+

+

+
Variable  Coefficient  Standard Error t

ratio P[T>t]  Mean of X
+

+

+

+

+

+

+
Constant

.13525584 .02375149

5.695 .0000
LOGY 1.00306313 .00296625 338.159 .0000 7.99083133
+

+

+

+

+

+

+
Newey

West Robust Covariance Matrix
Variable  Coefficient  Standard Error t

ratio P[T>t]  Mean of X
+

+

+

+

+

+

+
Constant

.13525584 .07257279

1.864 .0638
LOGY 1.00306313 .00938791 106.846 .0000 7.99083133
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FGLS
+

+
 AR(1) Model: e(t) = rho * e(t

1) + u(t) 
 Initial value of rho = .90693
 <<<***
 Maximum iterations = 100 

Method = Prais

Winsten

 Iter= 1, SS= .017, Log

L= 666.519353 
 Iter= 2, SS= .017, Log

L= 666.573544 
 Final value of Rho = .910496
 <<<***
 Iter= 2, SS= .017, Log

L= 666.573544 
 Durbin

Watson: e(t) = .179008 
 Std. Deviation: e(t) = .022308 
 Std. Deviation: u(t) = .009225 
 Durbin

Watson: u(t) = 2.512611 
 Autocorrelation: u(t) =

.256306 
 N[0,1] used for significance levels 
+

+
+

+

+

+

+

+

+
Variable  Coefficient  Standard Error b/St.Er.P[Z>z]  Mean of X
+

+

+

+

+

+

+
Constant

.08791441 .09678008

.908 .3637
LOGY .99749200 .01208806 82.519 .0000 7.99083133
RHO .91049600 .02902326 31.371 .0000
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Seemingly Unrelated Regressions
The classical regression model,
y
i
=
X
i
i
+
i
. Applies to
each of M equations and T observations. Familiar
example: The capital asset pricing model:
(
r
m

r
f
) =
m
i
+
m
(
r
market
–
r
f
) +
m
Not quite the same as a panel data model. M is usually
small

say 3 or 4. (The CAPM might have M in the
thousands, but it is a special case for other reasons.)
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Formulation
Consider an extension of the groupwise heteroscedastic
model: We had
y
i
=
X
i
+
i
with E[
i
X
] =
0,
Var[
i
X
] =
i
2
I
.
Now, allow two extensions:
Different coefficient vectors for each group,
Correlation across the observations at each specific
point in time (think about the CAPM above. Variation in
excess returns is affected both by firm specific factors
and by the economy as a whole).
Stack the equations to obtain a GR model.
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SUR Model
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OLS and GLS
Each equation can be fit by OLS ignoring all others. Why do GLS?
Efficiency improvement.
Gains to GLS:
None if identical regressors

NOTE THE CAPM ABOVE!
Implies that GLS is the same as OLS. This is an application of a
strange special case of the GR model. “If the K columns of
X
are
linear combinations of K characteristic vectors of
,楮it桥h䝒
model, then OLS is algebraically identical to GLS.” We will forego
our opportunity to prove this theorem. This is our only application.
(Kruskal’s Theorem)
Efficiency gains increase as the cross equation correlation increases
(of course!).
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The Identical
X
Case
Suppose the equations involve the same
X
matrices. (Not just the
same variables, the same data. Then GLS is the same as equation
by equation OLS.
Grunfeld’s investment data are not an example

each firm has its own
data matrix.
The 3 equation model on page 313 with Berndt and Wood’s data give
an example. The three share equations all have the constant and
logs of the price ratios on the RHS. Same variables, same years.
The CAPM is also an example.
(Note, because of the constraint in the B&W system (same
δ
parameters in more than one equation), the OLS result for identical
Xs does not apply.)
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Estimation by FGLS
Two step FGLS is essentially the same as the groupwise
heteroscedastic model.
(1) OLS for each equation produces residuals
e
i
.
(2)
S
ij
= (1/n)
e
i
e
j
then do FGLS
Maximum likelihood estimation for normally distributed
disturbances: Just iterate FLS.
(This is an application of the Oberhofer

Kmenta result.)
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Inference About the Coefficient Vectors
Usually based on Wald statistics.
If the estimator is maximum likelihood, LR statistic
T(log
S
restricted


log
S
unrestricted
)
is a chi

squared statistic with degrees of freedom
equal to the number of restrictions.
Equality of the coefficient vectors: (Historical note: Arnold Zellner, The
original developer of this model and estimation technique: “An
Efficient Method of Estimating Seemingly Unrelated Regressions
and
Tests of Aggregation Bias
”
(my emphasis). JASA, 1962, pp. 500

509.
What did he have in mind by “aggregation bias?”
How to test the hypothesis?
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Application
A Translog demand system
for a 3 factor process: (To bypass a
transition in the notation, we proceed directly to the application)
Electricity, Y, is produced using Fuel, F, capital, K, and Labor, L.
Theory:
The production function is Y = f(K,L,F). If it is smooth, has continuous
first and second derivatives, and if(1) factor prices are determined in
a market and (2) producers seek to minimize costs (maximize
profits), then there is a “cost function”
C = C(Y,PK,PL,PF).
Shephard’s Lemma states that the cost minimizing factor demands are
given by
Xm =
C(…)/
Pm.
Take logs gives the factor share equations,
logC(…)/
logPm = Pm/C
C(…)/
Pm = PmXm/C
which is the proportion of total cost spent on factor m.
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Translog
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Restrictions
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Data
–
C&G, N=123
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Ordinary least squares regression ............
LHS=C Mean =

.38339
Standard deviation = 1.53847
Number of observs. = 123
Model size Parameters = 10
Degrees of freedom = 113
Residuals Sum of squares = 2.32363
Standard error of e = .14340
Fit R

squared = .99195
Adjusted R

squared = .99131
Model test F[ 9, 113] (prob) = 1547.7(.0000)

+

Variable Coefficient Standard Error t

ratio P[T>t] Mean of X

+

Constant

7.79653 6.28338

1.241 .2172
Y .42610*** .14318 2.976 .0036 8.17947
YY .05606*** .00623 8.993 .0000 35.1125
PK 2.80754 2.11625 1.327 .1873 .88666
PL

.02630 (!)
2.54421

.010 .9918 5.58088
PKK .69161 .43475 1.591 .1144 .43747
PLL .10325 .51197 .202 .8405 15.6101
PKL

.48223 .41018

1.176 .2422 5.00507
YK

.07676** .03659

2.098 .0381 7.25281
YL .01473 .02888 .510 .6110 45.6830

+

Least Squares Estimate of Cost Function
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Criterion function for GLS is log

likelihood.
Iteration 0, GLS = 514.2530
Iteration 1, GLS = 519.8472
Iteration 2, GLS = 519.9199

Estimates for equation: C.........................
Generalized least squares regression ............
LHS=C Mean =

.38339
Residuals Sum of squares = 2.24766
Standard error of e = .14103
Fit R

squared = .99153
Adjusted R

squared = .99085
Model test F[ 9, 113] (prob) = 1469.3(.0000)

+

Variable Coefficient Standard Error b/St.Er. P[Z>z] Mean of X

+

Constant

9.51337** 4.26900

2.228 .0258
Y .48204*** .09725 4.956 .0000 8.17947
YY .04449*** .00423 10.521 .0000 35.1125
PK 2.48099* 1.43621 1.727 .0841 .88666
PL .61358 1.72652 .355 .7223 5.58088
PKK .65620** .29491 2.225 .0261 .43747
PLL

.03048 .34730

.088 .9301 15.6101
PKL

.42610 .27824

1.531 .1257 5.00507
YK

.06761*** .02482

2.724 .0064 7.25281
YL .01779 .01959 .908 .3640 45.6830

+

FGLS
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Constrained MLE for Multivariate Regression Model
First iteration: 0 F=

48.2305 logW=

7.72939 gtinv(H)g= 2.0977
Last iteration: 5 F= 508.8056 logW=

16.78689 gtinv(H)g= .0000
Number of observations used in estimation = 123
Model: ONE PK PL PKK PLL PKL Y YY YK YL
C B0 BK BL CKK CLL CKL CY CYY CYK CYL
SK BK CKK CKL CYK
SL BL CKL CLL CYL

+

Variable Coefficient Standard Error b/St.Er. P[Z>z
] (FGLS) (OLS)

+

B0

6.71218*** .21594

31.084 .
0000

9.51337

7.79653
CY .58239*** .02737 21.282 .0000
.48204 .42610
CYY .05016*** .00371 13.528 .0000
.04449 .05606
BK .22965*** .06757 3.399 .
0007
2.48099 2.80754
BL

.13562* .07948

1.706 .
0879 .61358

.02630
CKK .11603*** .01817 6.385 .
0000 .65620 .69161
CLL .07801*** .01563 4.991 .
0000

.03048 .10325
CKL

.01200 .01343

.894 .
3713

.42610

.48223
CYK


.00473* .00250

1.891 .
0586

.06761

.07676
CYL

.01792*** .00211

8.477 .
0000
.01779 .01473

+

Maximum Likelihood Estimates
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Vector Autoregression
The vector autoregression (VAR) model is one of the most successful, flexible,
and easy to use models for the analysis of multivariate time series. It is
a natural extension of the univariate autoregressive model to dynamic multivariate
time series. The VAR model has proven to be especially useful for
describing the dynamic behavior of economic and financial time series and
for forecasting. It often provides superior forecasts to those from univariate
time series models and elaborate theory

based simultaneous equations
models. Forecasts from VAR models are quite flexible because they can be
made conditional on the potential future paths of specified variables in the
model.
In addition to data description and forecasting, the VAR model is also
used for structural inference and policy analysis. In structural analysis, certain
assumptions about the causal structure of the data under investigation
are imposed, and the resulting causal impacts of unexpected shocks or
innovations to specified variables on the variables in the model are summarized.
These causal impacts are usually summarized with impulse response
functions and forecast error variance decompositions.
Eric Zivot: http://faculty.washington.edu/ezivot/econ584/notes/varModels.pdf
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VAR
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Zivot’s Data
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Impulse Responses
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